Meleshko V. Limit cycles and dissipative structures in systems described by differential equations with fractional derivatives

The thesis is devoted to mathematical modeling of limit cycles and dissipative structures in systems described by differential equations with fractional derivatives. This dissertation investigates the properties of nonlinear mathematical models with fractional derivatives, which are fundamental to the study of autowave processes in systems with integer derivatives. A linear stability analysis of basic autowave and autooscillation systems with fractional derivatives is performed. Using the examples of FitzHugh-Nagumo and Brusselator mathematical models, formation of limit cycles and dissipative structures in commensurate and incommensurate fractional systems is investigated. It is shown that the nonlinearities play a major role in shaping the structures and that the order of the fractional derivative is an additional bifurcation parameter that can change the stability conditions, the type of bifurcation and the nonlinear dynamic of system. It is presented that, depending on the value of fractional derivatives in basic systems there may be qualitatively different types of nonlinear autowave solutions. Using computer simulation to model systems, there has been constructed an area of instability, and the evolution of different types of autowave solutions has been studied. A detailed analysis of limit cycles and dissipative structures properties in the system has been performed. New types of nonlinear solutions in basic reaction-diffusion systems with fractional diffusion and fractional reaction are obtained. It is shown that in systems of this kind widely known types of autowave solutions and qualitatively new types of nonlinear dynamics system can exist. The properties of dissipative structures that can occur in stable areas in fractional reaction-diffusion systems are studied. Semi-analytical and numerical methods for the study of evolution of nonlinear systems with time fractional derivatives are proposed. The numerical algorithms and software packages for numerical simulation of nonlinear systems with fractional derivatives are developed. The nonlinear phenomena considered in this thesis are general in nature and can be used to investigate a wide class of nonlinear systems with fractional derivatives.